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Summary: JOURNAL OF COMBINATORIAL THEORY, Series A 40, 82-89 (1985)
An Extremal Problem for Sets with
Applications to Graph Theory
NOGA ALON*
Department of Mathematics, Massachusetts Institute of Technology,
Cambridge, Massachusetts 02139
Communicated b.vthe Managing Editors
Received December 28, 1983
Let X, ,...,X,, be n disjoint sets, For 1< i < n and 1
sets of X, that satisfy IA,I
(U,B,)=@ for l
h
Its proof uses multilinear techniques (exterior algebra). lrj 1985 Academic Press. Inc.
1. INTR~OLJOTI~N
Our main result in this paper is the following.
THEOREM 1.1. Let X, ,..., X, be n disjoint setsand let r1,..., r, and s,,..., s,
bepositive integers. For 1< id n and 1Qj < h let A, and B, besubsetsof Xi
that satisfy
lA,y < ri and IBijl
(v~+oB~)=@ for lGj
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